Finite Group Databasebeta

← #13362270

Subgroups — #13362270

Order 1792.

1126 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#442C211yesmaximal, characteristic
2C2×#89927C211yesmaximal, characteristic
3C4×#6499C212yesmaximal, complemented
4C2×#89790C212yesmaximal, complemented
5C2×#89488C211yesmaximal, characteristic, complemented
6#11729793C211yesmaximal, characteristic
7#11728329C211yesmaximal, characteristic, complemented
8#11728309C211yesmaximal, characteristic, complemented
9#11727946C212yesmaximal, complemented
10#11727912C212yesmaximal, complemented
11Dic7×#97C211yesmaximal, characteristic, complemented
12C22×C7×D8C2211yescharacteristic
13C2×C4×C7×D4C2212yes—
14C2×#6499C2212yes—
15C22×#1142C2212yes—
16C22×Dic28C2211yescharacteristic
17C2×C4×Dic14C2212yes—
18C7×#442—24no—
19C2×C7×#106C2211yescharacteristic
20C2×C7×#102C2212yes—
21C2×C7×#97C2211yescharacteristic
22#89993C2212yes—
23C2×#6406C2212yes—
24C4×C22⋊Dic7—24no—
25#89927C418yescomplemented
26C2×#6391C2211yescharacteristic
27C2×#6390C2212yes—
28C2×#6389C2211yescharacteristic
29C2×#6389C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89790—28no—
33#89749C2212yes—
34#89746C2212yes—
35C2×#6384C2212yes—
36C2×#6381C2212yes—
37#89717—24no—
38C42×Dic7C2211yescharacteristic
39C2×#6377C2211yescharacteristic
40#89488—24no—
41#21693—71nomaximal, Sylow
42C22×C7×D4C2312yes—
43C23×C4×C7—22no—
44C22×Dic14C2312yes—
45C2×C7×D8—22no—
46C2×C7×D8—22no—
47C2×C7×D8C2×C414yes—
48C22×C7×C8C2311yescharacteristic
49C22×C7×C8C2311yescharacteristic
50C4×C7×D4—24no—
51C4×C7×D4—24no—
52C2×C7×C4⋊C4C2312yes—
53C2×C7×C22⋊C4—22no—
54C2×C42×C7C2311yescharacteristic
55C2×C22⋊Dic7—24no—
56C2×C22⋊Dic7—24no—
57#6499—24no—
58#6499C2×C418yescomplemented
59C2×#1142—24no—
60C2×#1142—24no—
61C2×#1142C2×C418yescomplemented
62C2×C4⋊Dic7C2312yes—
63C2×Dic7⋊C4—22no—
64C2×C4×Dic7—22no—
65C2×C4×Dic7D414yes—
66C2×C4×Dic7C2311yescharacteristic
67C2×C4×Dic7C2311yescharacteristic
68C22×C7⋊C8C2311yescharacteristic
69C22×C7⋊C8C2311yescharacteristic
70C2×Dic28—22no—
71C2×Dic28—22no—
72C2×Dic28C2×C414yescomplemented
73C4×Dic14—24no—
74C4×Dic14—24no—
75C7×#106—22no—
76C7×#102—24no—
77C7×#97D414yes—
78#6406—24no—
79C7⋊#95—24no—
80C7⋊#95—24no—
81#6391—22no—
82#6390—24no—
83#6389—22no—
84#6389—22no—
85C8×Dic7—22no—
86C8×Dic7C2×C414yescomplemented
87#6384—24no—
88#6381—24no—
89#6377D414yes—
90C2×#495—71no—
91C2×#442D711yescharacteristic
92C2×#441—71no—
93C2×#440—72no—
94C4×#124—72no—
95#1973—72no—
96#1972—71no—
97#1971—71no—
98#1958—72no—
99#1846—71no—
100C4×#97—71no—
101C24×C7—22no—
102C2×C7×D4—22no—
103C2×C7×D4—22no—
104C2×C7×D4—22no—
105C2×C7×D4—22no—
106C2×C7×D4—22no—
107C2×C7×D4C22×C414yes—
108C22×C4×C7—22no—
109C22×C4×C7—48no—
110C22×C4×C7—44no—
111C22×C4×C7C2×D412yes—
112C22×C4×C7C2411yescharacteristic
113C22×Dic7—22no—
114C22×Dic7—22no—
115C22×Dic7C2×D412yes—
116C22×Dic7C2×D412yes—
117C2×Dic14—22no—
118C2×Dic14—22no—
119C2×Dic14—22no—
120C2×Dic14—22no—
121C2×Dic14—22no—
122C2×Dic14C22×C414yes—
123C7×D8—28no—
124C2×C7×C8—21no—
125C2×C7×C8—21no—
126C2×C7×C8C22×C412yes—
127C2×C7×C8C2×D412yes—
128C2×C7×C8C2×D412yes—
129C2×C7×C8#4112yes—
130C7×C4⋊C4—22no—
131C7×C4⋊C4—22no—
132C7×C22⋊C4—44no—
133C42×C7C2×D412yes—
134C42×C7C2×D412yes—
135C22⋊Dic7—216no—
136C22⋊Dic7—48no—
137#1142—216no—
138C4⋊Dic7—22no—
139C4⋊Dic7—22no—
140Dic7⋊C4—44no—
141C4×Dic7—21no—
142C4×Dic7—21no—
143C4×Dic7—44no—
144C4×Dic7—44no—
145C4×Dic7C22×C412yes—
146C4×Dic7#4112yes—
147C2×C7⋊C8—21no—
148C2×C7⋊C8—21no—
149C2×C7⋊C8C22×C412yes—
150C2×C7⋊C8C2×D412yes—
151C2×C7⋊C8C2×D412yes—
152C2×C7⋊C8#4112yes—
153Dic28—28no—
154C22×Q16—71no—
155C22×SD16—72no—
156C22×D8C2×D711yescharacteristic
157C2×#124—72no—
158C2×C4×Q8—72no—
159C2×C4×D4C2×D712yes—
160#495—78no—
161#442—24no—
162#441—144no—
163#440—148no—
164C2×#107—72no—
165C2×#106—71no—
166C2×#106C2×D711yescharacteristic
167C2×#103—72no—
168C2×#103—72no—
169C2×#102—72no—
170C2×#102C2×D712yes—
171C2×#97—71no—
172C2×#97—71no—
173C2×#97—71no—
174C2×#97C2×D711yescharacteristic
175C2×C4×C8—71no—
176C2×C4×C8—71no—
177#391—72no—
178#390—72no—
179#388—144no—
180#382—72no—
181C4×C22⋊C4—144no—
182C43—71no—
183C23×C7—24no—
184C23×C7—42no—
185C23×C7—42no—
186C23×C7—44no—
187C23×C7—44no—
188C23×C7C22×D411yescharacteristic
189C7×D4—24no—
190C7×D4—24no—
191C7×D4—42no—
192C7×D4—42no—
193C2×C4×C7—24no—
194C2×C4×C7—42no—
195C2×C4×C7—44no—
196C2×C4×C7—88no—
197C2×C4×C7#12714yes—
198C2×C4×C7C23×C411yescharacteristic
199C2×C4×C7C22×D411yescharacteristic
200C2×C4×C7C22×D411yescharacteristic
201C2×C4×C7C2×#4111yescharacteristic
202C2×C4×C7C2×#4111yescharacteristic
203C2×C4×C7C2×#4111yescharacteristic
204C2×Dic7—24no—
205C2×Dic7—24no—
206C2×Dic7—22no—
207C2×Dic7—24no—
208C2×Dic7—42no—
209C2×Dic7—42no—
210C2×Dic7—42no—
211C2×Dic7—44no—
212C2×Dic7C4×D414yes—
213Dic14—24no—
214Dic14—24no—
215Dic14—42no—
216Dic14—42no—
217C7×C8—22no—
218C7×C8C4×D414yes—
219C7⋊C8—22no—
220C7⋊C8C4×D414yes—
221C22×Q8—72no—
222C22×D4C22×D712yes—
223C23×C4—22no—
224C2×Q16—74no—
225C2×Q16—142no—
226C2×Q16—142no—
227C2×SD16—78no—
228C2×SD16—144no—
229C2×SD16—144no—
230C2×D8—22no—
231C2×D8—22no—
232C2×D8C4×D714yes—
233C22×C8—71no—
234C22×C8—71no—
235C22×C8C22×D711yescharacteristic
236C22×C8C22×D711yescharacteristic
237#124—78no—
238#124—144no—
239C4×Q8—144no—
240C4×Q8—144no—
241C4×D4—24no—
242C4×D4—24no—
243C2×C4⋊C4—72no—
244C2×C4⋊C4—142no—
245C2×C4⋊C4C22×D712yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—144no—
248C2×C22⋊C4—144no—
249C2×C42—71no—
250C2×C42—71no—
251C2×C42—74no—
252C2×C42—142no—
253C2×C42C22×D711yescharacteristic
254#107—144no—
255#106—22no—
256#106—142no—
257#103—144no—
258#103—144no—
259#102—24no—
260#102—144no—
261#97—74no—
262#97—142no—
263#97—142no—
264#97C22⋊D714yes—
265C4×C8—74no—
266C4×C8—142no—
267#95—144no—
268#95—144no—
269C22×C7—44no—
270C22×C7—48no—
271C22×C7—82no—
272C22×C7—82no—
273C22×C7—84no—
274C22×C7C2×C4×D411yescharacteristic
275C22×C7C2×#12711yescharacteristic
276C22×C7#53211yescharacteristic
277C22×C7#57514yes—
278C4×C7—24no—
279C4×C7—82no—
280C4×C7C2×C4×D411yescharacteristic
281C4×C7C2×C4×D411yescharacteristic
282C4×C7#53211yescharacteristic
283C4×C7#53211yescharacteristic
284Dic7—24no—
285Dic7—44no—
286Dic7—42no—
287Dic7—82no—
288C24—22no—
289C2×Q8—74no—
290C2×Q8—142no—
291C2×Q8—142no—
292C2×Q8—142no—
293C2×Q8—142no—
294C2×Q8—142no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4C2×C4×D714yes—
301C22×C4—22no—
302C22×C4—48no—
303C22×C4—44no—
304C22×C4—72no—
305C22×C4—72no—
306C22×C4—142no—
307C22×C4—142no—
308C22×C4D4×D712yes—
309C22×C4C23×D711yescharacteristic
310Q16—148no—
311SD16—1416no—
312D8—28no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—72no—
316C2×C8—72no—
317C2×C8—72no—
318C2×C8—72no—
319C2×C8—141no—
320C2×C8—141no—
321C2×C8C2×C4×D712yes—
322C2×C8#115712yes—
323C2×C8C2×C22⋊D712yes—
324C2×C8C2×C22⋊D712yes—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—142no—
328C4⋊C4—142no—
329C4⋊C4—284no—
330C22⋊C4—44no—
331C22⋊C4—1416no—
332C22⋊C4—288no—
333C42—72no—
334C42—72no—
335C42—141no—
336C42—141no—
337C42—284no—
338C42—284no—
339C42D4×D712yes—
340C42C2×C22⋊D712yes—
341C2×C7—84no—
342C2×C7C4×#12711yescharacteristic
343C2×C7#294312yes—
344C2×C7#315312yes—
345C2×C7#318312yes—
346C23—24no—
347C23—44no—
348C23—44no—
349C23—42no—
350C23—42no—
351C23C2×D4×D711yescharacteristic
352Q8—144no—
353Q8—144no—
354Q8—282no—
355Q8—282no—
356D4—24no—
357D4—24no—
358D4—42no—
359D4—42no—
360C2×C4—24no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—74no—
364C2×C4—88no—
365C2×C4—144no—
366C2×C4—144no—
367C2×C4—144no—
368C2×C4—142no—
369C2×C4—282no—
370C2×C4—282no—
371C2×C4—284no—
372C2×C4—282no—
373C2×C4#650314yes—
374C2×C4C22×C4×D711yescharacteristic
375C2×C4C2×#115711yescharacteristic
376C2×C4C2×D4×D711yescharacteristic
377C2×C4D7×#4111yescharacteristic
378C2×C4D7×#4111yescharacteristic
379C2×C4C22×C22⋊D711yescharacteristic
380C8—22no—
381C8—74no—
382C8—142no—
383C8C4×C22⋊D714yes—
384C7#2169311yescharacteristic, complemented, Sylow
385C22—44no—
386C22—48no—
387C22—82no—
388C22—82no—
389C22—84no—
390C22C4×D4×D711yescharacteristic
391C22#9023911yescharacteristic
392C22#9046414yes—
393C22C2×#650311yescharacteristic
394C4—24no—
395C4—82no—
396C4—144no—
397C4—284no—
398C4—282no—
399C4—562no—
400C4C4×D4×D711yescharacteristic
401C4#9023911yescharacteristic
402C4C2×C4×C22⋊D711yescharacteristic
403C4#9052311yescharacteristic
404C2—84no—
405C2#1173730712yes—
406C2#1173730912yes—
407C2C4×#650311yescharacteristic
408C2#1174138312yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#442C2
socleC23×C7C22×D4
prime radicalC22×C7×D8C22
cosocleC22×C4×C7C24
hypercenterC2×#442D7
hypocenterC7#21693
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C4×#6499C2semidirect
4. C2×#89790C2semidirect
5. C2×#89488C2semidirect
7. #11728329C2semidirect
8. #11728309C2semidirect
9. #11727946C2semidirect
10. #11727912C2semidirect
11. Dic7×#97C2semidirect
25. #89927C4semidirect
58. #6499C2×C4semidirect
61. C2×#1142C2×C4semidirect
72. C2×Dic28C2×C4semidirect
86. C8×Dic7C2×C4semidirect
384. C7#21693semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C7×#4422
C2×#899272
C4×#64992
C2×#897902
C2×#894882
#117297932
#117283292
#117283092
#117279462
#117279122
Dic7×#972
#216937